University of Virginia Library

MATHEMATICS IN
CULTURAL HISTORY

The history of mathematics is but a thin ribbon across
the fabric of general history, that is, of the cultural
history of events, insights, and ideas. Yet there are
major problems of general history that are meaning-
fully refracted in the history of mathematics, and one
such problem, on which we shall concentrate, is the
problem of explaining the decline of ancient civili-
zation in the West. On the other hand, it may happen
that mathematics is materially involved in a problem
in general history, and yet cannot contribute to its
illumination; one such problem, on which we intend
to comment, is the problem of the rise and spread of

phonetic writing. Finally, there are intriguing situa-
tions in which the judgment of mathematics on the
eminence of an era is different from that of other fields
of knowledge, even of physics proper, and in several
brief preliminary sections we shall rapidly review a
few problems of this kind from periods beginning with
and following upon the Renaissance.

But before beginning our reviews we wish to state
that the problems to be encountered will be formulated
in broad, summary, and even simplistic terms. This is
to be expected. As a mode of rational cognition mathe-
matics appears rather early and is quite central, but
as a mode of intellectual activity it is rather primitive.
For this reason mathematics is effective because of its
strength rather than its delicacy, even when involved
in sensibilities. Therefore, when participating in the
analysis of problems from general history, the history
of mathematics is at its best when the problems are
stated in large, manifest, even crude fashion rather than
in a localized, esoteric, and delicate manner.

The Renaissance. There is a much-studied problem
whether and in what sense there indeed was a Renais-
sance—in the fifteenth and sixteenth centuries, say—as
institutionalized by Jacob Burckhardt; that is, whether
there is a marked-off era which interposes itself be-
tween medieval and modern times. (For a history of
the problems see W. K. Ferguson, K. H. Dannenfeldt,
T. Helton, also E. Panofsky.) George Sarton, the leading
historian of science, once made the statement (which
he later greatly modified) that “from the scientific point
of view, the Renaissance was not a renaissance” (quoted
in Dannenfeldt, p. 115). Of course, nobody would deny
that there was a Copernicus in the sixteenth century
or that the introduction of printing created a great
spurt in many compartments of knowledge, scientific
or other; but the question, which Sarton's statement
was intended to answer, is whether during the era of
the Renaissance, disciplines like physics, chemistry,
biology, geology, economics, history, philosophy, etc.,
were developed in a manner which sets these two
centuries off from both the Middle Ages and the seven-
teenth century.

Now, with regard to mathematics no such doubts
need, or even can arise. There was indeed a mathe-
matics of the Renaissance that was original and dis-
tinctive in its drives and characteristics. Firstly, there
was a school, mainly, but not exclusively, represented
by Germans (Peurbach, Regiomontanus, and others),
that sharply advanced the use of symbols and notations
in arithmetic and algebra. Secondly, and strikingly, an
Italian school sharply advanced the cause of the alge-
bra of polynomial equations when it solved equations
of the third and fourth degree in terms of radicals
(Scipione del Ferro, Ludovico Ferrari, Nicolo Tartag
lia, Geronimo Cardano). Thirdly, a French school,
mainly represented by François Viète, achieved a syn-
thesis of these two developments. And finally, an “in-
ternational” school laid the foundation for the eventual
rise of analysis by introducing and studying two special
classes of functions, trigonometric functions and loga-
rithms. Regiomontanus, Rheticus, Johann Werner, and
later Viète gradually made trigonometry an inde-
pendent part of mathematics, and Henry Briggs and
John Napier (and perhaps also Jost Bürger) created the
logarithmic (and hence also exponential) functions.
After that, for over three centuries trigonometric and
logarithmic functions were the stepping-stones leading
to the realm of analysis, for students of mathematics
on all levels.

There is a near-consensus among social historians
that the rise of arithmetic and algebra during the
Renaissance was motivated “preponderantly by the rise
of commerce in the later Middle Ages; so that the
challenges to which the rise of algebra was the response
were predominantly the very unlofty and utilitarian
demands of counting houses of bankers and merchants
in Lombardy, Northern Europe and the Levant”
(Bochner, Role..., p. 38). On this explanation, the
socioeconomic needs that were thus satisfied were not
those of the “industrialist” but those of the merchant,
in keeping with the fact that in the late Middle Ages,
and soon after, the general economy was dominated
not by the producer but by the trader (ibid.).

The Seventeenth Century. There was one thing that
the mathematics of the Renaissance era did not
achieve. It did not continue creatively the mathematics
of Euclid, Archimedes, and Apollonius, even if it did
begin to translate and study their works. The exploita-
tion in depth of Greek mathematics was achieved only
in the seventeenth century, in the era of the Scientific
Revolution. It is this development that created the
infinitesimal calculus, and thus molded the world image
which we have inherited today. For instance, Johannes
Kepler anticipated the infinitesimal calculus in two
ways; by his approach to Archimedean calculations of
volumes, and also by his manner of reasoning which
led him to conclude that planetary orbits are not circles
but ellipses; and without his intimate knowledge of
Apollonius this conclusion would have hardly come
about. Next, René Descartes' avowed aim in his La
Géométrie was to solve or re-solve geometrical prob-
lems of the great Greek commentator Pappus (third
century A.D.) who was groping for topics and attitudes
in geometry that were beyond his pale. Now, Des-
cartes' technical equipment was the operational appa-
ratus of Viète, and it was this fusion of Viète with
Pappus that created our coordinate and algebraic ge-
ometry. Next, Isaac Barrow, the teacher of Isaac

Newton, perceived the importance of Euclid's 5th book
(theory of proportion in lieu of our theory of positive
real numbers) for the eventual unfolding of analysis.
And, finally, Newton himself, by an extraordinary
tour-de-force, pressed his forward-directed Principia
(Principles of Natural Philosophy) into the obstructive
mold of backward-directed but powerful Archimedism.
That this impressment was even harder on Newton's
readers than on Newton is attested to by a memorable
statement (in the nineteenth century) of William
Whewell:

(Whewell, I, 408).

To sum it up, it was not the Renaissance proper of
the fifteenth and sixteenth centuries that saw the
“rebirth” of antiquity as much as it was the succeeding
Scientific Revolution of the seventeenth century. What
the Renaissance did was to contribute a prerequisite
algebraic symbolism, and this was something that, from
our retrospect, antiquity might have additionally pro-
duced out of itself but did not.

The Age of Enlightenment. The merits and achieve-
ments of this age are subject to most diverse and diver-
gent interpretations. For instance, Carl Becker, in a
small but unforgettable book, The Heavenly City of
the Eighteenth-century Philosophers (1932), proposed
that, on the whole, the intellectual attitudes of the age
were medieval rather than modern, Voltaire or no
Voltaire. Notwithstanding assertions by some historians
of science, the advancement in physical science was
rather circumscribed. For instance, the eighteenth
century did not achieve much in the theory of elec-
tricity, at any rate not before Henry Cavendish and
C. A. Coulomb, who were active towards the end of
the century after the sheen of the Enlightenment had
already been dimmed. Also, optics virtually stood still
for over a century. After the lustrous works of Newton,
Christiaan Huygens, and others in the seventeenth
century almost nothing memorable happened in optics
till the early part of the nineteenth century when
Thomas Young, Étienne-Louis Malus, Augustin Fresnel,
William Hyde Wollaston, and Joseph von Fraunhofer
began to crowd the field. Furthermore, Immanuel
Kant's famous Critiques came toward the end of the
eighteenth century, after the “true” Enlightenment had
begun to fade; and Kant's predecessors in metaphysics
earlier in the century had been not a whit better than
Kant had depicted them.

But the eminence of mathematics in the era of
Enlightenment is clear-cut. It was a very great century.
Mathematicians like G. W. Leibniz, Jacob and John
Bernoulli, A. C. Clairaut, L. Euler, J. le R. D'Alembert,
P. L. M. de Maupertuis, J. L. Lagrange, and P. S. de
Laplace made it as distinctive a century as any since
Pythagoras among the Greeks, or even since the age
of Hammurabi in Mesopotamia. Also the age had one
feature that made it simply unique. It fused mathe-
matics and mechanics in a manner and to a degree
that were unparalleled in any other era, before or after.
Also, in mathematics, far from being a “medieval” age
as in Carl Becker's conception of the eighteenth cen-
tury, it was a very “modern” age. Monumental as
Newton's Principia (1686) may have been, it is Lag-
range's Mécanique analytique (1786) that became the
basic textbook of our later physical theory. Lagrange's
treatise is old-fashioned, but readable, Newton's treatise
is “immortal,” but antiquarian, and very difficult.

It is not easy to follow in depth the growth of this
mathematics in relation to other developments of the
era. Socioeconomic motivations do not account for its
high level, and there are no explanations from general
philosophy that are convincing. Immanuel Kant, for
instance, shows no familiarity at all with the advanced
mathematics of his times.

The Nineteenth Century. In the history of knowl-
edge, even more complex than the era of Enlighten-
ment is an era following it. It is, schematically, the
half-century centered in the year 1800, that is the
half-century 1776-1825, which has been called the Age
of Eclosion by Bochner. During this era there was a
great and sudden outburst of knowledge in all academic
disciplines; and in historical studies of various kinds,
there emerged a “critical” approach to the evaluation
of source material, which is sometimes called “higher
criticism.” In mathematics there was no such sudden
increase of activity, but an analogue to the “higher
criticism” did begin to manifest itself. A critical ap-
proach to so-called foundations of mathematics began
to spread, and even novel patterns of insight were
emerging. Thus, C. F. Gauss in his Disquisitiones arith-
meticae, which appeared in 1801, explicitly formulated
the statement that any integer can be represented as
a product of prime numbers, and uniquely so. Implic-
itly the statement is already contained, between the
lines, in Book 7 of Euclid's Elements. But explicitly
the statement is a kind of “existence” and “uniqueness”
theorem, for which even number theorists like Fermat,
Wallis, Euler, and Lagrange were “not ready” yet.

Afterwards, in the course of the nineteenth century,
mathematics filled all those “gaps” in Greek mathe-
matics, which the Greeks themselves could not or
would not close. Thus, the nineteenth century finally

elucidated the role of Euclid's axiom on parallels, and
the role of axioms in general, as well as in geometry;
and it finally constructed a “Euclidean” space as a
background space for (Greek) mathematical figures and
astronomical orbits. In a peculiar logical sense, Greek
mathematics was “completed” only around 1900, but
by the developments which brought this about, it was
also rendered “antiquarian” by them. On the other
hand, “humanistic” Greek works like those of Homer
and Aeschylus, Herodotus and Thucydides, and Plato
and Aristotle were not “completed” in a similar sense,
but they have, on the other hand, not been rendered
“antiquarian,” as shelves full of reissues of these works
in paperback can attest.

This completes our preliminary observations, and we
now turn to two topics from antiquity for a somewhat
less summary analysis.

Phonetic Writing. We are taking it for granted that
within the history of civilization there is a correlation,
an important one, between the rise of organized
knowledge and the emergence of adequate writing, the
degree of adequacy being measured by the degree of
phonetic articulation. With a suitable definition of
“phonetic,” we may say that Chinese writing is a fully
developed phonetic system, and has been so since its
appearance about the middle of the second millennium
B.C. (Gelb, p. 85). Now, mathematics is the oldest
organized knowledge there is, or nearly so, and our
problem is the task of assessing the role of mathematics
in this correlation between knowledge and writing.

On hard evidence, the presence of organized mathe-
matics is first attested around 1800 B.C. Now, our first
intricacy is the fact that this first evidence for organ-
ized mathematics appears in two separate areas simul-
taneously, in Mesopotamia and in Egypt (Neugebauer,
Chs. 1-3). This simultaneity of appearance cannot be
readily explained by invoking a “cultural diffusion” or
only a “stimulus diffusion” because by intent and con-
tent these two mathematics are very different from
each other (Neugebauer, loc. cit.; van der Waerden,
Chs. 1-3). Both these systems of mathematics are
documented by writing that is highly organized (see
Gelb, pp. 168ff., for Egypt, and Kramer, p. 306, for
Mesopotamia), but of seemingly different provenance.
In both geographical areas the mathematics in use
appears to be a full part of organized knowledge in
general. Also, from our retrospect, the Mesopotamian
mathematics even gives the impression of having been
in the very vanguard of organized knowledge, although
to an “average Mesopotamian” of the era, the code
of Hammurabi may have been more important, and,
above all, much more familiar.

At this point it might be expected that if writing
and knowledge reach a certain stage of organization
conjointly, then organized mathematics would do so
too. But this would be a very hasty expectation, as can
be seen from the development in an area that geo-
graphically and intellectually was very proximate to
both Mesopotamia and Egypt, namely, the land of the
Bible, where writing of various modes was organizing
itself in the latter half of the second millennium B.C.,
and in this area it was even advancing towards its
ultimate stage, namely, to the stage of becoming en-
tirely alphabetical (Gelb, pp. 134-53). At the same
time, juridical, ethical, and sacerdotal knowledge was
organizing itself too, and much of it became ultimately
knowledge for the ages. Yet, the history of mathematics
knows absolutely nothing about an indigenous mathe-
matics also springing up in this area at the same time.
This is our most serious intricacy.

A different problem arises—and it is a research task
rather than a conundrum—when we take into account
Greek achievements, from the first millennium B.C. It
is a fact that the Greeks made writing fully alphabeti-
cal. They thus created “a writing which expresses the
single sounds of a language” (Gelb, p. 197), and their
script was undoubtedly more advanced than Babylon-
ian script from around 1800 B.C. Similarly, the mathe-
matics which the Greeks began shaping within their
own thought patterns in the sixth century B.C. was,
even from the first, undoubtedly more advanced than
the Babylonian mathematics from around 1800 B.C.
Now, the research task arising is to determine whether
these two advances are commensurate in extent. Greek
mathematics drew, as heavily as it could, on all the
accumulated mathematics that preceded it. Never-
theless, it is immediately clear that the Greek intellec-
tual innovation in organized mathematics—as also in
organized knowledge of any kind—was far greater than
the parallel advance in the phonetic quality of writing.
An analysis in depth might counteract this impression.
But such an analysis could not be an easy one, because
it would also have to account for the fact that modern
mathematics is immeasurably superior to the Greek
creation, although “from the Greek period up to the
present nothing has happened in the inner structural
development of writing” (Gelb, p. 184).

Finally, we wish to observe that both for organized
mathematics and organized writing it is equally diffi-
cult to decide whether in China either came into being
independently of the West, or in direct dependence
on the West, or by a combination of the two possi-
bilities at different times. For organized mathematics
the problem of its geographic propagation in Asia was
already known, more or less, in the nineteenth century,
perhaps under the impact of the corresponding prob-
lem for language as it is spoken. In the twentieth
century the problem for mathematics has received less

attention than the corresponding problem for writing,
early or organized, probably because for mathematics
the problem has been less able to exploit achievements
in archeology than those in writing. For instance, for
writing there is a balanced account of the problem in
Gelb, Ch. VIII, but there seems to be no analogous
account for mathematics.

Dissolution of Ancient Civilization. By “ancient
civilization” we mean, as usual, the large conglom-
eration of component civilizations of so-called antiq-
uity that severally came into being in and around the
Mediterranean Littoral. In the last centuries B.C. and
first centuries A.D., the conglomeration merged into one
compound civilization, creating “one civilized world,”
the self-styled oikouméne. The passing-away of this
civilization constitutes one of history's greatest prob-
lems. The central part of this problem is the seemingly
“formal” question of determining when ancient civili-
zation terminated, if indeed it did “terminate”; or,
rather, since the termination was probably a process
of some duration, when the process of termination
began and when it ended.

It is widely accepted that the process ended towards
the close of the fifth century A.D., as symbolized by
the fact that in 476 A.D. “the last claimant to the
Roman throne in the West was deposed” (Kagan, p.
viii). An almost lone dissenter among historians, but
a leading one, is Henri Pirenne, who maintains that
ancient civilization was brought to an end—and then
rather violently so—only in the seventh century A.D.,
by the widespread militancy of Islam. Ancient civili-
zation was then not actually “killed,” but rather pushed
away from its Mediterranean habitat into more north-
ern parts of Europe, where it became isolated and
immobilized for centuries (Havighurst, 1958).

Another kind of dissent, a qualified one, is caused
inevitably by the fact that there was an Eastern Roman
Empire which lasted very long, until 1453 A.D. Most
Byzantinists of today find that this empire was very
viable in 476 A.D. and beyond, until the onset of the
Crusades at any rate. They do not deny that there was
a ruin of the Western Empire in the fifth century, but
most of them do not allow that this ruin came about,
to a meaningful degree, by decay from within. They
argue that the causes for decay from within would have
been present also in the Eastern half of the Empire,
and should have had the same destructive effect at the
time (Jones, Ch. 24; J. B. Bury, quoted in Kagan, pp.
7-10), so that, in their view, only military and other
external causes remain. A very outspoken champion
of this view is Lynn White, Jr., and he is severely
critical of the details, attitudes, and emphases in
Edward Gibbon's key work, Decline and Fall of the
Roman Empire (White, pp. 291-311).

With regard to the beginning of the process of the
decline of ancient civilization widely spaced dates have
been proposed, explicitly or by implication. F. W.
Walbank finds “... the germs of the illness of antiquity
already present in the Athens of the fifth century B.C.”
(Kagan, p. viii). This is a very early date indeed. Less
extreme is the finding of M. I. Rostovtzeff that “decline
began as early as the second century B.C.” (Kagan, p.
2). Finally “according to Gibbon, the Roman Empire
reached its zenith in the age of the Antonines [second
century A.D.] after which the decline set in” (White,
p. 25). Thus, by implication, in the view of Gibbon
the decline set in around 200 A.D., after the era of the
“Five Good Emperors” (90-180 A.D.).

We proceed now to review the cogency of the above
dates in the light of the history of mathematics.

The history of mathematics fully corroborates the
familiar textbook assertion that around 500 A.D. a
large-scale decline occurred. Mathematics as an intel-
lectual activity—as an academic discipline, so to
speak—was suddenly lost from sight, as if swallowed
up by a wave of a flood, or buried by a sandstorm.
In the Latinized West there had been a mathematics
bearing the telltale mark of a Greek heritage even
when dealing with non-Greek or extra-Greek topics
of the mathematical corpus. It was this kind of mathe-
matics that suddenly disappeared. Also in the West,
this kind of mathematics came fully to surface only
after seven centuries or so, in the famous Liber abaci
of Leonardo of Pisa (Fibonacci) around 1200 A.D. This
is not to say that in the intervening centuries mathe-
matics was unknown in the West. A mathematics of
a kind was of course included in school curricula. There
were some translations from the Arabic, especially
during the so-called twelfth-century Renaissance.
There was also a certain pursuit of “utilitarian” math-
ematics, even during the so-called Carolingian Renais-
sance (Smith, I, 175-220). But, before the work of
Fibonacci, this pursuit did not evince a quest for the
kind of originality, if only on a modest scale, with
which mathematics had been imbued since the sixth
century B.C., when the Greeks had begun to weave
mathematics into the texture of their rationality.

This decay of Greek mathematics did not spare the
Eastern Roman empire, which, by an official reckoning,
lasted from 529 to 1453, from Justinian to the fall of
Constantinople. As far as mathematics is concerned this
Empire might have never been. There is an encyclo-
pedic treatise which lists the extant Byzantine writings,
including works on mathematics and cognate subjects
(Krumbacher, pp. 620-26). The mathematical works
of the collection bespeak sterility and stagnation, and
only in the art of warfare were there some elements
of originality (Taton, p. 446).

Finally, we note that, in the judgment of a leading
student of chronological innovations (Ginzel, III,
178ff.), at the beginning of the sixth century A.D. the
Roman Canonist Dionysius Exiguus founded our
present-day system of designating years by A.D. and
B.C. This event, even if conceived very modestly by
its author, was, from our retrospect, mathematically
tinged, and it took place in Rome, after its “official”
fall, even if not long after it. Thus, this may be viewed
as a mathematical corroboration of the fact—which
was stressed, from very different approaches, by Henri
Pirenne, Lynn White, and probably others—that the
political fall of Rome in 476 A.D. was not, instantly,
also a social and intellectual disintegration.

This achievement of the Roman Canonist, modest
as it may appear to be when viewed in isolation, cannot
be overestimated as a determinant of history. The
Greeks never quite succeeded in introducing a com-
parable dating of years. Mathematically this would
have amounted to introducing a coordinate system on
the time axis, and the Greeks never achieved this.
Furthermore, it is remarkable that the “Christian” era
of Dionysius not only became a “common” era but
has turned out to be the most durable era ever. The
French Revolution of 1793, the Russian Revolution of
1917, the Italian Revolution of 1922, and the German
Revolution of 1933, each attempted to introduce a new
era beginning with itself, but none really succeeded
or even made the attempt in earnest. Our “common”
era, however “Christian” by origin, has become a
standard institution that cannot be tampered with.

We now turn to the question of mathematical evi-
dence for the beginning of the decline of ancient civi-
lization. Firstly, mathematics clearly concurs with the
assertion of Rostovtzeff that a general decay began in
the second century B.C. Secondly, mathematics can
offer no tangible corroboration of the fact, known from
general history, that life in the Greco-Roman world
was much bleaker in the third century A.D. than in
the preceding one. Thirdly, and finally, mathematics
can even corroborate the thesis of Walbank that germs
of some of the illnesses of antiquity can already be
found in the Athens of the fifth century B.C.; namely,
by a tour-de-force, we may elicit from the nature of
Greek mathematics some peculiar comment of the
thesis, which can be interpreted as a corroboration of
it, in a sense.

Greek mathematics built on a considerable body of
mathematics that had preceded it, but it was never-
theless a singular achievement of ancient civilization
as a whole, and a hallmark of its Hellenic aspect in
particular. Naively or boldly, the Greeks made a fresh
start. They were inspired to begin from a new begin-
ning and they succeeded. They erected an edifice of
mathematics that was a veritable “system” in our
present-day sense of the word. The bricks and stones
in the edifice may have been Egyptian, Babylonian,
or other, but the structure was Greek. This Greek
mathematics attained its intellectual acme in the
achievements of Archimedes. Isaac Newton even com-
posed his Principia in an Archimedean mise-en-scène,
but he acknowledged no indebtedness to an Egyptian
calculus of fractions, or even a Babylonian calculus of
quadratic equations, even if he knew anything about
them at all.

From what is known, this Greek mathematics
showed the first signs of being itself around 600 B.C.
It then grew and kept unfolding for about four centur-
ies, that is, till about 200 B.C., and the last of these
four centuries, that is the era from 300 B.C. to 200 B.C.,
was a culminating one. In fact, around 300 B.C. Euclid
composed his Elements, Archimedes flourished around
250 B.C., and around 200 B.C., Apollonius produced his
monumental Conics.

But after that, in the second century B.C., unexpect-
edly and inexplicably, as if on a signal, the development
of this mathematics came almost to a halt. After 200
B.C. it began to level off, to loose its impetus, and then
to falter, bringing to the fore only such works as those
of a Nicomedes, Dioclos, and Hypsicles. The phenom-
enon was no passing setback but a permanent recession,
the beginning of a decline. It was, for mathematics,
the beginning of the end in the conception of Ros-
tovtzeff, even if the true end itself, that is the final
extinction of Greek mathematics in its own phase,
came only considerably later, around 500 A.D., that is,
around the time of the fall of Rome.

It is noteworthy though, that the second century B.C.,
as if to almost show that it was not entirely down and
out, produced the astronomer Hipparchus, famed dis-
coverer of the precession of the equinoxes; and that
the second century A.D., as if to lay claim to being
indeed a “good” century, brought forth his great “suc-
cessor” Claudius Ptolemy, author of the majestic
Almagest, and of a Geography. It must be stated how-
ever, emphatically, that in “basic” mathematics
Ptolemy was in no wise farther along than Archimedes,
even if the Almagest, as an astronomical text, was a
live text still for Copernicus in the first half of the
sixteenth century and began to become antiquarian
only half a century later in consequence of the mathe-
matically articulated innovations of Kepler.

Limitations of Greek Mathematics. The reasons that
have been variously adduced for the dissolution of
ancient civilization—the overextension of the oikou-
méne so that “the stupendous fabric yielded to the
pressure of its own weight”; inadequacy of industriali-
zation and too much involvement with slave labor;

declining manpower; loss of economic freedom; “the
gradual absorption of the educated classes by the
masses”; “the pitiful poverty of Western Rome”; etc.
(Kagan, pp. xi and xii)—may all help to account for
the ultimate extinction of Greek mathematics, around
500 A.D., after a gradual decline of long duration. One
may even add the view of J. B. Bury that “the gradual
collapse... was the consequence of a series of con-
tigent events. No general cause can be assigned that
made it inevitable” (ibid.).

But, in the case of mathematics there is one peculiar
fact which no such reasons from general history can
really account for. It is the fact that in the second
century B.C., much before the ultimate extinction, the
decline of mathematics from the heights which it had
attained in the preceding century was seemingly too
large, too brusque, and too unmotivated by internal
developments to be satisfactorily explained by general
reasons of this kind. By standard criteria of advance-
ment, mathematics in the third century B.C. was in a
state of upward development, and it suggests itself that
the rather sudden break in the development after 200
B.C. may have been due, at least in part, to some
particular reasons applying to mathematics only. This
is indeed our suggestion, and we shall attempt to for-
mulate it.

In the third century B.C., Greek mathematics was
not only very good, but it also reached a climax. By
this we mean that it reached a level of development
that was maximal relative to the intellectual base,
mathematical and philosophical, on which it had been
erected and on which it rested. Therefore, mathematics
could have continued to develop in the second century
B.C. and later only if the overall intellectual base on
which it rested could also have been broadened in the
process. But of this kind of broadening of the total
intellectual setting of mathematics, Greek civilization
in the second century B.C. was no longer capable. The
general intellectual basis for Greek mathematics, which
in a sense never broadened or deepened, was laid in
the sixth and fifth centuries B.C., especially the latter,
and in this peculiarly conceived sense it can be said
that, as far as mathematics is concerned the decline
of Greek civilization reaches back even into the fifth
century B.C. (Walbank).

In order to demonstrate that the mathematics of
Archimedes and Apollonius was overripe relative to
its intellectual basis we shall compare the conceptual
setting in Archimedes and Apollonius with the corre-
sponding setting in Newton's Principia (1686), even if
Newton's work came nineteen centuries later. A com-
parison of the works of Archimedes and Apollonius
(and Pappus) with La Géométrie (1637) of Descartes,
which was published half a century before the Prin
cipia, would not serve our present purpose, because
Descartes does not retain the setting of antiquity. On
the contrary, he radically changed the technical setting
by a full recourse to the apparatus of algebraic sym-
bolism as made ready for him by Viète. Not so Newton.
He was most expert in the handling of this apparatus,
and on occasions he employed it more penetratingly
than Descartes and others; but, for reasons best known
to himself he elected to cast the Principia in a mold
of Archimedean technicalities, outwardly, that is. In
the Principia there are hardly any analytical formulas;
but there are circumlocutions and verbalized formulae
which, at times, seem to be as condensed and stero-
typed as in Archimedes. This makes for hard reading
nowadays, but it makes it easy to isolate differences
of approach and setting. The differences are enormous,
and we list the following ones.

Newton prominently introduced an underlying
overall space, his absolute space, as a background space
for both mathematics and mechanics. The Greeks
achieved nothing like it. They certainly did not intro-
duce a space for mechanics and mathematics jointly.
They did introduce a “place” for events in nature
which perhaps served as a space of mechanics, but they
most certainly did not ever introduce a space of math-
ematics, or any kind of space of perception, physical,
logical, or ontological. In mathematics, they had “loci”
for individual figures when constructed, but not a space
for such figures before being constructed. In short, the
Greeks did not have any kind of space in the sense
of Descartes, or Newton, or John Locke.

Newton expressly introduced in his mechanics a
translational momentum (quantity of motion), defining
it, for a mass particle moving on a straight line, as
the product m · v in which the factor m is the constant
amount of mass of the particle and v is its instantaneous
velocity. Archimedes, in his theory of the lever, ought
to have introduced the conception of a rotational mo-
mentum, defining it as the product l · p in which the
factor l is the length of an arm of the lever and p is
the weight suspended from this arm. But Archimedes
did not introduce such a concept, nor did Greek math-
ematical thought ever conceptualize a product like
l · p; and mechanics went on marking time for almost
2000 years.

Even more significantly, Newton had the concept
of a function constantly in his thinking, however cov-
ertly. Altogether since the seventeenth century the
concept of a function kept on occurring in many facets
and contexts, in mathematics as well as in other areas
of cognition. Greek cognition, however, never had the
notion of function, anywhere. Even the absence of
products like l · p from Greek thinking was part of the
absence of functions, inasmuch as in mathematics of

today a product l · p for variable values of l and p,
is a function on the set of pairs (l,p). More centrally,
in cognition today the most important component of
the concept of function is the notion of relation, how-
ever elusive it may be, to define or even describe what
a relation is. Aristotle, the creator of the academic
discipline of logic, did not anticipate the importance
of relation (which he terms pros ti), nor did the Stoic
logicians after him. But in modern developments, the
creation of an algebra or logic by the American
philosopher-logician Charles Sanders Peirce was his
most outstanding logical achievement.

Operationally, functions occur in Newton's Principia
in the following way. If a mass particle moves on the
x-axis and t denotes the time variable, then Newton
covertly assumes that there is a function x(t) which is
the instantaneous distance of the particle from a fixed
origin. He forms the derivative dx/dt for variable t,
which is a new function v = v(t). It is the instantaneous
velocity of the motion. He multiplies this by the con-
stant value m of the mass, thus introducing the instan-
taneous quantity of motion m · v(t), which is again a
function in the variable t. Newton then crowns these
covert assumptions with the hypothesis, which is ap-
parently due to himself, that the force F which brings
about the motion is, at every instant, equal to the rate
of change of the quantity of motion, F = d(mv)/dt.

This hypothesis, coupled with Newton's specific law
of gravitation, created our exact science of today. The
Greeks did not conceive of any part of this entire
context of assumptions and hypothesis, not because
they were unable to form a derivative of a function,
but because they did not have in their thinking the
concept of a function that is a prerequisite to forming
the various derivatives involved. By maturity of insight,
Archimedes was better equipped than Newton to carry
out the limit process that is involved in the formation
of a derivative, if only the concept of function and
the entire prerequisite setting had been given to him.

The Greek lack of familiarity with the concept of
function does not manifest itself only in mathematical
mechanics, which, to the Greeks was a relatively eso-
teric topic, but also in the entire area of geometry,
which, by a common conception, was a stronghold of
Greek rationality. There is a purely geometrical con-
text, common to Archimedes and Newton, in which
Newton does, and Archimedes does not have functions
in his thinking. Namely, Newton views the tangent to
a curve at a point of the curve as the limiting position
of a secant through the fixed point and a variable point
of the curve, so that, in effect, he performs the opera-
tion of differentiation on “hidden” coordinate func-
tions. Greek mathematics, however, never broke
through to this all-important view, but persisted in the
view, known from Euclid, that a tangent to a curve
is a straight line which in its entire extent coincides
with the curve at one point only. Archimedes tries to
adhere to this Euclidean definition even in his essay
on (Archimedean) spirals, in spite of the complication,
of which he is apparently aware, that any straight line
in the plane of the spiral intersects it in more than
one point. Without putting it into words, Archimedes
overcomes the complication by a simple adjustment,
but he does not advance towards the modern concep-
tion of a tangent as in Newton.

Epilogue. In modern mathematics the Greek limita-
tions which we have adduced were overcome mainly
by conceptual innovations, namely by the creation of
abstractions, and of escalations of abstractions, which
do not conform with the cognitive texture of Greek
classical philosophy and general knowledge. There is
an all-pervading difference between modern mathe-
matical abstractions and, say, Platonic ideas; reductions
of the one to the other, as frequently attempted in
philosophy of mathematics, are forced and unconvinc-
ing. There are analogies and parallels between the two,
but not assimilations and subordinations. The Greeks
could form the (Platonic) idea of a “general” triangle,
quadrangle, pentagon, and even of a “general” poly-
gon, but the conception of a background space for
Euclid's geometry was somehow no longer such an idea
and eluded them. A Platonic idea, even in its most
“idealistic” form, was still somehow object-bound,
which a background space for mathematics no longer
is. Nor could the Greeks form the “idea” of a rotational
momentum, for it simply is no longer an “idea,” and
cannot be pressed into the mold of one. It is, quanti-
tatively, a product l · p in which l and p represent
“ideally” heterogenous objects, but are nevertheless
measured by the same kind of positive real number;
and real numbers themselves are already abstractions
pressing beyond the confines of mere “ideas.” Such
fusion of several abstractions into one was more than
the Greek could cope with; and the formation of
(Newton's) translational momentum, as presented skel-
etally above, was even farther beyond their intellectual
horizon.

If it is granted that Greek mathematics has been thus
circumscribed, it becomes a major task of the history
of ideas—and not only of the history of mathe-
matics—to determine by what stages of medieval de-
velopment, gradual or spontaneous—the inherited
mathematics was eventually made receptive to sym-
bolic and conceptual innovations during the Renais-
sance and after.

This task is inseparable from the task of determining
the originality and effectiveness of medieval Arabic
knowledge, mathematical and other, and its durable

influence on the Latinized West. Only within this kind
of setting will it be possible to comprehend the course
of mathematics in its conceptual and cultural aspects.

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SALOMON BOCHNER

[See also Infinity; Mathematical Rigor; Newton on Method;
Relativity; Renaissance Humanism; Space; Symmetry.]